Tree of Heaven (Ailanthus altissima)
Identification
Tree of Heaven can grow up for 70 feet or more, but the vast majority you’ll see around town are saplings 5 feet and under. They have compound leaves with oppositely arranged leaflets (each leaflet is directly across the stem from another leaflet), except for the leading leaflet which points straight ahead. This leads to it often being confused with the Black Walnut or Staghorn Sumac, who have similar leaf arrangement.
The telltale sign to identify a Tree of Heaven is the small lobes at the bottom of each leaflet, illustrated below. If you see any of these, it is definitely a Tree of Heaven.
Removal
The primary concern when removing Tree of Heaven is it’s incredible suckering ability. This means that when the tree becomes stressed, the roots will panic and send up new trees all over the place. Even if you rip the the tree out of the ground, if any of the root remains, there is a good chance that suckers (new trees) will grow from those roots. This means that the level of root development will determine the best course of action for removal.
Small sapling (less than a foot tall)
If it's a this small, it's roots probably haven't established enough to sucker even if you do leave some behind when pulling it up. You can just yank this out like any weed, and even if it does sucker, it you can just yank those too and it'll be over with pretty quick.
Larger sapling
You'll want to be more thorough. Get as much root out of the ground as you can, and be prepared to be pulling suckers out of the ground in the near future. Depending on root development and your thoroughness, this could be 1 to dozens of suckers. Eventually the roots will starve if you keep yanking the suckers.
Established tree
Dealing with an established Tree of Heaven means tricking it into thinking its not dying an imminent death. Cutting it down or killing it quick will result in a plantation of new Trees of Heaven surrounding the old stump pretty quick.
The preferred method here would be to knick and paint. This involves using small amount of herbicide on a paintbrush. Do not ever spray herbicide, the collateral damage from herbicide spray is immense. You should pick a spot on the trunk, maybe 3 or 4 feet off the ground, and cut small vertical strips into the trunk every 6-12 inches or so. So on a smaller tree you might only have 2 cuts, but a larger tree might have a handful or more. Paint each of these with a small amount of Triclopyr and leave it be. Do it again every 6 months or so until the tree dies, at which point it can be felled. This should, hopefully, not alarm the tree to its imminent death and this prevent major suckering from the roots.